A cyclic quadrilateral $ABCD$ has diameter $AC$ with circumcircle $\omega$. Let $K$ be the foot of the perpendicular from $C$ to $BD$, and the tangent to $\omega$ at $A$ meets $BD$ at $T$. Let the line $AK$ meets $\omega$ at $X$ and choose a point $Y$ on line $AK$ such that $\angle TYA=90^{\circ}$. Prove that $AY=KX$. Proposed by Anzo Teh Zhao Yang

A finite sequence of decimal digits from $\{0,1,\cdots, 9\}$ is said to be common if for each sufficiently large positive integer $n$, there exists a positive integer $m$ such that the expansion of $n$ in base $m$ ends with this sequence of digits. For example, $0$ is common because for any large $n$, the expansion of $n$ in base $n$ is $10$, whereas $00$ is not common because for any squarefree $n$, the expansion of $n$ in any base cannot end with $00$. Determine all common sequences. Proposed by Wong Jer Ren

Given $n$ students in the plane such that the $\frac{n(n-1)}{2}$ distances are pairwise distinct. Each student gives a candy each to the $k$ students closest to him. Given that each student receives the same amount of candies, determine all possible values of $n$ in terms of $k$. Proposed by Wong Jer Ren

Fix a real polynomial $P$ with degree at least $1$, and a real number $c$. Prove that there exist a real number $k$ such that for all reals $a$ and $b$, $$P(a)+P(b)=c \quad \Rightarrow \quad |a+b|<k$$ Proposed by Wong Jer Ren

Do there exist infinitely many positive integers $a, b$ such that $$(a^2+1)(b^2+1)((a+b)^2+1)$$is a perfect square? Proposed Ivan Chan Guan Yu

Let $n$ be a positive integer, and Megavan has a $(3n+1)\times (3n+1)$ board. All squares, except one, are tiled by non-overlapping $1\times 3$ triominoes. In each step, he can choose a triomino that is untouched in the step right before it, and then shift this triomino horizontally or vertically by one square, as long as the triominoes remain non-overlapping after this move. Show that there exist some $k$, such that after $k$ moves Megavan can no longer make any valid moves irregardless of the initial configuration, and find the smallest possible $k$ for each $n$. (Note: While he cannot undo a move immediately before the current step, he may still choose to move a triomino that has already been moved at least two steps before.) Proposed by Ivan Chan Kai Chin

Let $P$ be the set of all primes. Given any positive integer $n$, define $$\displaystyle f(n) = \max_{p \in P}v_p(n)$$Prove that for any positive integer $k\ge 2$, there exists infinitely many positive integers $m$ such that \[ f(m+1) = f(m+2) = \cdots = f(m+k) \] Proposed by Ivan Chan Guan Yu

Given a triangle $ABC$, let $I$ be the incenter, and $J$ be the $A$-excenter. A line $\ell$ through $A$ perpendicular to $BC$ intersect the lines $BI$, $CI$, $BJ$, $CJ$ at $P$, $Q$, $R$, $S$ respectively. Suppose the angle bisector of $\angle BAC$ meet $BC$ at $K$, and $L$ is a point such that $AL$ is a diameter in $(ABC)$. Prove that the line $KL$, $\ell$, and the line through the centers of circles $(IPQ)$ and $(JRS)$, are concurrent. Proposed by Chuah Jia Herng & Ivan Chan Kai Chin